The kinetic energy K of a body executing simple harmonic motion is plotted against time, the time being expressed in terms of the period T. The initial conditions are such that the phase angle is zero. Such a graph would most closely resemble which of the following graphs?
A) 1
B) 2
C) 3
D) 4
E) 5

Question

The kinetic energy K of a body executing simple harmonic motion is plotted against time, the time being expressed in terms of the period T. The initial conditions are such that the phase angle is zero. Such a graph would most closely resemble which of the following graphs? 
A) 1 
B) 2 
C) 3 
D) 4 
E) 5

Quizplus · Accepted Answer

The kinetic energy of a simple harmonic oscillator is given by K = (1/2)mv^2 = (1/2)kA^2cos^2(ωt). At the equilibrium position, the velocity is zero, so the kinetic energy is also zero. As the body starts moving, it gains some kinetic energy, which reaches a maximum at the extremes of its motion and zero at the turning points. Therefore, the graph of kinetic energy vs time should resemble the graph of the displacement of a simple harmonic oscillator. The graph of displacement vs time for a simple harmonic oscillator is sinusoidal, with zeros at integer multiples of the period T. The graph of kinetic energy vs time should have its maxima at the same points where the displacement is zero, and its minima at the same points where the displacement is maximum. Only graph C satisfies these conditions, as it shows an oscillatory behavior with maxima and minima at integer multiples of T. Graphs A, B, D, and E do not resemble the behavior of a simple harmonic oscillator, and therefore are not plausible choices for the graph of K vs T.

The Kinetic Energy K of a Body Executing Simple Harmonic